Comparing Arrays

Last week in Math Workshop, we spent time arranging different amounts of chairs into rows and columns (arrays). Today, we are comparing the arrangement 16 and 17 chairs. Look at the different arrangements that we can make with each number below:

What do you notice about these arrays? Do any of the arrays have a special or unique shape? What do you notice about the number of arrays that can be made with 16 chairs compared to 17 chairs?

Hopefully, you notice that 16 chairs can be arranged in several different arrays. That is because it has many factors: 1, 2, 4, 8, 16. A number that has more than two factors is called a composite number. You probably noticed that 17 only has two arrays. This is because 17 is a prime number. Any number that has only two factors, one and itself, is a prime number. The factors of 17 are 17 and 1.

Also, you may have noticed that 16 can be arranged into a perfect square with 4 rows and 4 columns. Any number that results when another number is multiplied by itself is a square number. (ex: 3x3=9 Nine is a square number. 5x5=25 Twenty-five is a square number.) Sometimes math vocabulary can be confusing! For a reminder of the meaning of some math words that you may have forgotten, visit this great online math dictionary.

Thank you to our guest author this week, Miss Russell. :)

Fractions of Arrays

What amount of the array below is red?

To figure this out, we need to talk about fractions. Now that we have become experts with arrays, columns, and rows, we have moved into separating our arrays into fractional parts. Time for a quick fraction review?
A fraction is a part of a whole. The bottom number of a fractions is called the denominator. It tells you the size of the units that you have. The top number of a fraction is called the numerator. It tells you how many units you have. In the example to the left, the one half tells us that there are two units. One of the units is red.
We have been using wholes, halves, thirds, and fourths in creating our rectangular arrays. Students use square tiles to create these arrays in class. To recreate this at home, simply cut out the squares of the one-inch graph paper and arrange them in different arrays. Let's look at some examples below.

Figure A: This figure shows one whole. The entire array is green, so we say that one whole of the array is green.

Figure B: This figure shows one half. Three pieces of the array are red and three pieces of the array are green. The array is broken into two equal colors, so we say that one half of the array is red and one half of the array is green.

Figure C: This figure shows thirds. Two pieces of the array are blue, two pieces of the array are green, two pieces of the array are red. The array is broken into three equal colors, so we say that one third of the array is blue, one third is green, and one third is red.

Figure D: This figure shows fourths. Two pieces of the array are yellow, two pieces of the array are green, two pieces of the array are red, two pieces of the array are blue. The array is broken into four equal colors, so we say that one fourth of the array is yellow, one fourth is green, one fourth is red, and one fourth is blue.

What are fractions? Kind-friendly explanation.

What is 1 whole? Kid-friendly explanation.

What fraction is shown? Game

Match the Fractions Game

Ed Helper Fractions You do need a subscription to this site to access most of the practice sheets, however the subscription is well worth it! (You can also have access to practice reading comprehension tests, vocabulary and language practice, science, social studies, etc. with the subscription.)

Rectangles 101

Time to get back to the basics of rectangles. A rectangle is a quadrilateral (4 sides) with 4 right angles. Students were given 7 rectangles of various sizes in class and asked to sort them in order from least to greatest. As always, all manipulatives and materials were available for students. Once the mad rush for rulers ended, I began hearing students arguing in their small groups about the order of the rectangles. Some put them in order by height, some by width, some by “eyeballing” the size. Then, I overheard a group ask “What about the stuff on the inside of the rectangle? Some of them have more room on the inside.” And thus the area conversation was born.

We use square tile manipulatives to determine the area of shapes in second grade. Students began covering the rectangles with tiles. They were then able to put the rectangles in order from least to greatest by counting the tiles on the rectangles.

This caused another problem. Students had to figure out a way to describe what they did so that another group could recreate it. In class, when we try to recreate something, we say that we are making congruent shapes. (We are making shapes that are the same exact shape and same exact size as the shape described.) We discovered that this was impossible to do without using a common vocabulary.

We began to look at our rectangles as arrays. An array is simply an arrangement of rows and columns. We determined that we would call the vertical parts or our array columns and the horizontal parts rows. We were then all finally able to put the rectangles in order from least to greatest.

We are still new to rectangles, so it may take a little practice to master columns and rows. I was not able to find much online by way of rectangle games. I am attaching 1 inch graph paper to this post to use to print and practice at home.

You can use this paper several different ways:
-Cut out the squares and use them to create arrays to practice counting the areas. Play different games such as giving your child only a certain amount of squares to use for each rectangle and seeing how many different rectangles they can make. You could also tell your child the number of columns and rows and have your child recreate the array.
-Draw different arrays on the graph paper and count the number of columns and rows.
-Cut out several large rectangles and have your child practice putting them in order from least to greatest.

Online practice finding area.